01/15/2008, 12:41 AM
About GFR's "yellow zone" (YZ) or Ivars' "phase transition region" I was very interested to see the formula \( x = (y^{-y} -1)\frac{e^{-e}}{e^{1/e}-1} \). I think it is a very good approximation, but I think it is not accurate. In the YZ, the variables you gave should satisfy \( y = x^{x^y} \) for each of the two stems off the main graph. This equation has been discussed by both Knoebel ("Exponentials Reiterated"), and Galidakis ("... solving the p-th auxiliary equation..." or something), so we should probably see what they say about the subject. Until then I am including a graph which includes (black = GFR-approximation, blue = super-roots of height 10000/10001).